I have a question concerning conditional independence. According to wikipedia (yes, maybe not the best source) two random variables are conditionally independent given a third if $$p(x,y|z) = p(x|z)p(y|z)\:\:\forall z.$$
However, I read a mutual information definition of conditional independence that said that two random variables are conditionally independent if $$I[X|Z:Y|Z] = \left\langle\left\langle\log \frac{p(x,y|z)}{p(x|z)p(y|z)}\right\rangle_{X,Y|Z}\right\rangle_Z = 0.$$
Now, I am not sure whether the two equations are equivalent. The first definitely implies the last. However, the last does not imply the first (or does it?), since there could be $z$ for which $p(x,y|z) \not= p(x|z)p(y|z)$ but the set of all $z$ for which that is true has measure zero.
I guess the last equation means conditionally independence only almost surely (in terms of $z$) but not pointwise, but the first requires pointwise equality.
Am I correct? If not, where is my mistake?